Wave–particle duality is the concept that every elementary particle or quantic entity exhibits the properties of not only particles, but also waves. It addresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantumscale objects. As Einstein wrote: "It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do".^{[1]}
Various opinions have arisen about this.
Initiated by Louis de Broglie, before the discovery of quantum mechanics, and developed later as the de BroglieBohm theory, the pilot wave interpretation does not regard the duality as paradoxical, seeing both particle and wave aspects as always coexisting. According to Schrödinger, the domain of the de Broglie waves is ordinary physical spacetime.^{[2]} This formal feature in principle makes an account separable in ordinary physical spacetime. It thereby serves to exhibit this particular theory as nonlocal, which is considered by many physicists to be a grave defect in a theory.^{[3]}
Still in the days of the old quantum theory, another prequantummechanical version of wave–particle duality was pioneered by William Duane,^{[4]} and developed by others including Alfred Landé.^{[5]} Duane explained how a crystal diffracts of Xrays in terms solely of their particle aspect. The deflection of the trajectory of each diffracted photon was due to quantal translative momentum transfer from the spatially regular structure of the diffracting crystal.^{[6]} Fourier analysis reveals the wave–particle duality as a simply mathematical equivalence, always present, and universal for all quanta. The same reasoning applies for example to diffraction of electrons by a crystal.
In the light of de Broglie's ideas, Erwin Schrödinger developed his wave mechanics by referring the universal wave aspect not to ordinary physical spacetime, but rather to a profoundly different and more abstract 'space'. The domain of Schrödinger's wave function is configuration space.^{[2]} Ordinary physical spacetime allows more or less direct visualization of cause and effect relations, and so is said to be separable. In that sense, configuration space is not separable, and does not directly show cause and effect linkages. Sometimes, nevertheless, it seemed as if Schrödinger visualized his own waves as referring to ordinary spacetime, and there was much debate about this.^{[7]}
Niels Bohr regarded the "duality paradox" as a fundamental or metaphysical fact of nature. Sometimes the wave aspect was apparent, and sometimes the particle aspect, with the same kind of quantic entity, but in respectively different physical settings. He saw it as one aspect of the concept of complementarity.^{[8]}^{:242, 375–376} Bohr regarded renunciation of the causeeffect relation, or complementarily, of the spacetime picture, as essential to the quantum mechanical account.^{[9]}
Werner Heisenberg considered the question further. He saw the duality as present for all quantic entities, but not quite in the usual quantum mechanical account considered by Bohr. He saw it in what is called second quantization, which generates an entirely new concept of fields which exist in ordinary spacetime, causality still being visualizable. Classical field values (e.g. the electric and magnetic field strengths of Maxwell) are replaced by an entirely new kind of field value, as considered in quantum field theory. Turning the reasoning around, ordinary quantum mechanics can be deduced as a specialized consequence of quantum field theory.^{[10]}^{[11]}
Because of the difference of views of Bohr and Heisenberg, the main sources of the socalled Copenhagen interpretation, the position of that interpretation on wave–particle duality is illdefined.
Origin of theory
The idea of duality originated in a debate over the nature of light and matter that dates back to the 17th century, when Christiaan Huygens and Isaac Newton proposed competing theories of light: light was thought either to consist of waves (Huygens) or of particles (Newton). Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, and many others, current scientific theory holds that all particles also have a wave nature (and vice versa).^{[12]} This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. For macroscopic particles, because of their extremely short wavelengths, wave properties usually cannot be detected.^{[13]}
Brief history of wave and particle viewpoints
Aristotle was one of the first to publicly hypothesize about the nature of light, proposing that light is a disturbance in the element air (that is, it is a wavelike phenomenon). On the other hand, Democritus—the original atomist—argued that all things in the universe, including light, are composed of indivisible subcomponents (light being some form of solar atom).^{[14]} At the beginning of the 11th Century, the Arabic scientist Alhazen wrote the first comprehensive treatise on optics; describing refraction, reflection, and the operation of a pinhole lens via rays of light traveling from the point of emission to the eye. He asserted that these rays were composed of particles of light. In 1630, René Descartes popularized and accredited the opposing wave description in his treatise on light, showing that the behavior of light could be recreated by modeling wavelike disturbances in a universal medium ("plenum"). Beginning in 1670 and progressing over three decades, Isaac Newton developed and championed his corpuscular hypothesis, arguing that the perfectly straight lines of reflection demonstrated light's particle nature; only particles could travel in such straight lines. He explained refraction by positing that particles of light accelerated laterally upon entering a denser medium. Around the same time, Newton's contemporaries Robert Hooke and Christiaan Huygens—and later AugustinJean Fresnel—mathematically refined the wave viewpoint, showing that if light traveled at different speeds in different media (such as water and air), refraction could be easily explained as the mediumdependent propagation of light waves. The resulting Huygens–Fresnel principle was extremely successful at reproducing light's behavior and, subsequently supported by Thomas Young's 1803 discovery of doubleslit interference, was the beginning of the end for the particle light camp.^{[15]}^{[16]}
Thomas Young's sketch of twoslit diffraction of waves, 1803
The final blow against corpuscular theory came when James Clerk Maxwell discovered that he could combine four simple equations, which had been previously discovered, along with a slight modification to describe selfpropagating waves of oscillating electric and magnetic fields. When the propagation speed of these electromagnetic waves was calculated, the speed of light fell out. It quickly became apparent that visible light, ultraviolet light, and infrared light (phenomena thought previously to be unrelated) were all electromagnetic waves of differing frequency. The wave theory had prevailed—or at least it seemed to.
While the 19th century had seen the success of the wave theory at describing light, it had also witnessed the rise of the atomic theory at describing matter. In 1789, Antoine Lavoisier securely differentiated chemistry from alchemy by introducing rigor and precision into his laboratory techniques; allowing him to deduce the conservation of mass and categorize many new chemical elements and compounds. However, the nature of these essential chemical elements remained unknown. In 1799, Joseph Louis Proust advanced chemistry towards the atom by showing that elements combined in definite proportions. This led John Dalton to resurrect Democritus' atom in 1803, when he proposed that elements were invisible sub components; which explained why the varying oxides of metals (e.g. stannous oxide and cassiterite, SnO and SnO_{2} respectively) possess a 1:2 ratio of oxygen to one another. But Dalton and other chemists of the time had not considered that some elements occur in monatomic form (like Helium) and others in diatomic form (like Hydrogen), or that water was H_{2}O, not the simpler and more intuitive HO—thus the atomic weights presented at the time were varied and often incorrect. Additionally, the formation of H_{2}O by two parts of hydrogen gas and one part of oxygen gas would require an atom of oxygen to split in half (or two halfatoms of hydrogen to come together). This problem was solved by Amedeo Avogadro, who studied the reacting volumes of gases as they formed liquids and solids. By postulating that equal volumes of elemental gas contain an equal number of atoms, he was able to show that H_{2}O was formed from two parts H_{2} and one part O_{2}. By discovering diatomic gases, Avogadro completed the basic atomic theory, allowing the correct molecular formulae of most known compounds—as well as the correct weights of atoms—to be deduced and categorized in a consistent manner. The final stroke in classical atomic theory came when Dimitri Mendeleev saw an order in recurring chemical properties, and created a table presenting the elements in unprecedented order and symmetry. But there were holes in Mendeleev's table, with no element to fill them in. His critics initially cited this as a fatal flaw, but were silenced when new elements were discovered that perfectly fit into these holes. The success of the periodic table effectively converted any remaining opposition to atomic theory; even though no single atom had ever been observed in the laboratory, chemistry was now an atomic science.
Animation showing the waveparticle duality with a double slit experiment and effect of an observer. Increase size to see explanations in the video itself. See also
quiz based on this animation.
Turn of the 20th century and the paradigm shift
Particles of electricity
At the close of the 19th century, the reductionism of atomic theory began to advance into the atom itself; determining, through physics, the nature of the atom and the operation of chemical reactions. Electricity, first thought to be a fluid, was now understood to consist of particles called electrons. This was first demonstrated by J. J. Thomson in 1897 when, using a cathode ray tube, he found that an electrical charge would travel across a vacuum (which would possess infinite resistance in classical theory). Since the vacuum offered no medium for an electric fluid to travel, this discovery could only be explained via a particle carrying a negative charge and moving through the vacuum. This electron flew in the face of classical electrodynamics, which had successfully treated electricity as a fluid for many years (leading to the invention of batteries, electric motors, dynamos, and arc lamps). More importantly, the intimate relation between electric charge and electromagnetism had been well documented following the discoveries of Michael Faraday and James Clerk Maxwell. Since electromagnetism was known to be a wave generated by a changing electric or magnetic field (a continuous, wavelike entity itself) an atomic/particle description of electricity and charge was a non sequitur. Furthermore, classical electrodynamics was not the only classical theory rendered incomplete.
Radiation quantization
Blackbody radiation, the emission of electromagnetic energy due to an object's heat, could not be explained from classical arguments alone. The equipartition theorem of classical mechanics, the basis of all classical thermodynamic theories, stated that an object's energy is partitioned equally among the object's vibrational modes. This worked well when describing thermal objects, whose vibrational modes were defined as the speeds of their constituent atoms, and the speed distribution derived from egalitarian partitioning of these vibrational modes closely matched experimental results. Speeds much higher than the average speed were suppressed by the fact that kinetic energy is quadratic—doubling the speed requires four times the energy—thus the number of atoms occupying high energy modes (high speeds) quickly drops off because the constant, equal partition can excite successively fewer atoms. Low speed modes would ostensibly dominate the distribution, since low speed modes would require ever less energy, and prima facie a zerospeed mode would require zero energy and its energy partition would contain an infinite number of atoms. But this would only occur in the absence of atomic interaction; when collisions are allowed, the low speed modes are immediately suppressed by jostling from the higher energy atoms, exciting them to higher energy modes. An equilibrium is swiftly reached where most atoms occupy a speed proportional to the temperature of the object (thus defining temperature as the average kinetic energy of the object).
But applying the same reasoning to the electromagnetic emission of such a thermal object was not so successful. It had been long known that thermal objects emit light. Hot metal glows red, and upon further heating, white (this is the underlying principle of the incandescent bulb). Since light was known to be waves of electromagnetism, physicists hoped to describe this emission via classical laws. This became known as the black body problem. Since the equipartition theorem worked so well in describing the vibrational modes of the thermal object itself, it was trivial to assume that it would perform equally well in describing the radiative emission of such objects. But a problem quickly arose when determining the vibrational modes of light. To simplify the problem (by limiting the vibrational modes) a longest allowable wavelength was defined by placing the thermal object in a cavity. Any electromagnetic mode at equilibrium (i.e. any standing wave) could only exist if it used the walls of the cavities as nodes. Thus there were no waves/modes with a wavelength larger than twice the length (L) of the cavity.
Standing waves in a cavity
The first few allowable modes would therefore have wavelengths of : 2L, L, 2L/3, L/2, etc. (each successive wavelength adding one node to the wave). However, while the wavelength could never exceed 2L, there was no such limit on decreasing the wavelength, and adding nodes to reduce the wavelength could proceed ad infinitum. Suddenly it became apparent that the short wavelength modes completely dominated the distribution, since ever shorter wavelength modes could be crammed into the cavity. If each mode received an equal partition of energy, the short wavelength modes would consume all the energy. This became clear when plotting the Rayleigh–Jeans law which, while correctly predicting the intensity of long wavelength emissions, predicted infinite total energy as the intensity diverges to infinity for short wavelengths. This became known as the ultraviolet catastrophe.
The solution arrived in 1900 when Max Planck hypothesized that the frequency of light emitted by the black body depended on the frequency of the oscillator that emitted it, and the energy of these oscillators increased linearly with frequency (according to his constant h, where E = hν). This was not an unsound proposal considering that macroscopic oscillators operate similarly: when studying five simple harmonic oscillators of equal amplitude but different frequency, the oscillator with the highest frequency possesses the highest energy (though this relationship is not linear like Planck's). By demanding that highfrequency light must be emitted by an oscillator of equal frequency, and further requiring that this oscillator occupy higher energy than one of a lesser frequency, Planck avoided any catastrophe; giving an equal partition to highfrequency oscillators produced successively fewer oscillators and less emitted light. And as in the Maxwell–Boltzmann distribution, the lowfrequency, lowenergy oscillators were suppressed by the onslaught of thermal jiggling from higher energy oscillators, which necessarily increased their energy and frequency.
The most revolutionary aspect of Planck's treatment of the black body is that it inherently relies on an integer number of oscillators in thermal equilibrium with the electromagnetic field. These oscillators give their entire energy to the electromagnetic field, creating a quantum of light, as often as they are excited by the electromagnetic field, absorbing a quantum of light and beginning to oscillate at the corresponding frequency. Planck had intentionally created an atomic theory of the black body, but had unintentionally generated an atomic theory of light, where the black body never generates quanta of light at a given frequency with an energy less than hν. However, once realizing that he had quantized the electromagnetic field, he denounced particles of light as a limitation of his approximation, not a property of reality.
Photoelectric effect illuminated
Yet while Planck had solved the ultraviolet catastrophe by using atoms and a quantized electromagnetic field, most physicists immediately agreed that Planck's "light quanta" were unavoidable flaws in his model. A more complete derivation of black body radiation would produce a fully continuous, fully wavelike electromagnetic field with no quantization. However, in 1905 Albert Einstein took Planck's black body model in itself and saw a wonderful solution to another outstanding problem of the day: the photoelectric effect, the phenomenon where electrons are emitted from atoms when they absorb energy from light. Ever since the discovery of electrons eight years previously, electrons had been the thing to study in physics laboratories worldwide.
In 1902 Philipp Lenard discovered that (within the range of the experimental parameters he was using) the energy of these ejected electrons did not depend on the intensity of the incoming light, but on its frequency. So if one shines a little lowfrequency light upon a metal, a few low energy electrons are ejected. If one now shines a very intense beam of lowfrequency light upon the same metal, a whole slew of electrons are ejected; however they possess the same low energy, there are merely more of them. In order to get high energy electrons, one must illuminate the metal with highfrequency light. The more light there is, the more electrons are ejected. Like blackbody radiation, this was at odds with a theory invoking continuous transfer of energy between radiation and matter. However, it can still be explained using a fully classical description of light, as long as matter is quantum mechanical in nature.^{[17]}
If one used Planck's energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy quantum hν, then the photoelectric effect could be explained very simply. Lowfrequency light only ejects lowenergy electrons because each electron is excited by the absorption of a single photon. Increasing the intensity of the lowfrequency light (increasing the number of photons) only increases the number of excited electrons, not their energy, because the energy of each photon remains low. Only by increasing the frequency of the light, and thus increasing the energy of the photons, can one eject electrons with higher energy. Thus, using Planck's constant h to determine the energy of the photons based upon their frequency, the energy of ejected electrons should also increase linearly with frequency; the gradient of the line being Planck's constant. These results were not confirmed until 1915, when Robert Andrews Millikan, who had previously determined the charge of the electron, produced experimental results in perfect accord with Einstein's predictions. While the energy of ejected electrons reflected Planck's constant, the existence of photons was not explicitly proven until the discovery of the photon antibunching effect, of which a modern experiment can be performed in undergraduatelevel labs.^{[18]} This phenomenon could only be explained via photons, and not through any semiclassical theory (which could alternatively explain the photoelectric effect). When Einstein received his Nobel Prize in 1921, it was not for his more difficult and mathematically laborious special and general relativity, but for the simple, yet totally revolutionary, suggestion of quantized light. Einstein's "light quanta" would not be called photons until 1925, but even in 1905 they represented the quintessential example of waveparticle duality. Electromagnetic radiation propagates following linear wave equations, but can only be emitted or absorbed as discrete elements, thus acting as a wave and a particle simultaneously.
Developmental milestones
Huygens and Newton
The earliest comprehensive theory of light was advanced by Christiaan Huygens, who proposed a wave theory of light, and in particular demonstrated how waves might interfere to form a wavefront, propagating in a straight line. However, the theory had difficulties in other matters, and was soon overshadowed by Isaac Newton's corpuscular theory of light. That is, Newton proposed that light consisted of small particles, with which he could easily explain the phenomenon of reflection. With considerably more difficulty, he could also explain refraction through a lens, and the splitting of sunlight into a rainbow by a prism. Newton's particle viewpoint went essentially unchallenged for over a century.^{[19]}
Young, Fresnel, and Maxwell
In the early 19th century, the doubleslit experiments by Young and Fresnel provided evidence for Huygens' wave theories. The doubleslit experiments showed that when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves; the wavelength of light can be computed from such patterns. The wave view did not immediately displace the ray and particle view, but began to dominate scientific thinking about light in the mid 19th century, since it could explain polarization phenomena that the alternatives could not.^{[20]}
In the late 19th century, James Clerk Maxwell explained light as the propagation of electromagnetic waves according to the Maxwell equations. These equations were verified by experiment by Heinrich Hertz in 1887, and the wave theory became widely accepted.
Planck's formula for blackbody radiation
In 1901, Max Planck published an analysis that succeeded in reproducing the observed spectrum of light emitted by a glowing object. To accomplish this, Planck had to make an ad hoc mathematical assumption of quantized energy of the oscillators (atoms of the black body) that emit radiation. It was Einstein who later proposed that it is the electromagnetic radiation itself that is quantized, and not the energy of radiating atoms.
Einstein's explanation of the photoelectric effect
The photoelectric effect. Incoming photons on the left strike a metal plate (bottom), and eject electrons, depicted as flying off to the right.
In 1905, Albert Einstein provided an explanation of the photoelectric effect, a hitherto troubling experiment that the wave theory of light seemed incapable of explaining. He did so by postulating the existence of photons, quanta of light energy with particulate qualities.
In the photoelectric effect, it was observed that shining a light on certain metals would lead to an electric current in a circuit. Presumably, the light was knocking electrons out of the metal, causing current to flow. However, using the case of potassium as an example, it was also observed that while a dim blue light was enough to cause a current, even the strongest, brightest red light available with the technology of the time caused no current at all. According to the classical theory of light and matter, the strength or amplitude of a light wave was in proportion to its brightness: a bright light should have been easily strong enough to create a large current. Yet, oddly, this was not so.
Einstein explained this conundrum by postulating that the electrons can receive energy from electromagnetic field only in discrete portions (quanta that were called photons): an amount of energy E that was related to the frequency f of the light by

E = h f\,
where h is Planck's constant (6.626 × 10^{−34} J seconds). Only photons of a high enough frequency (above a certain threshold value) could knock an electron free. For example, photons of blue light had sufficient energy to free an electron from the metal, but photons of red light did not. More intense light above the threshold frequency could release more electrons, but no amount of light (using technology available at the time) below the threshold frequency could release an electron. To "violate" this law would require extremely high intensity lasers which had not yet been invented. Intensitydependent phenomena have now been studied in detail with such lasers.^{[21]}
Einstein was awarded the Nobel Prize in Physics in 1921 for his discovery of the law of the photoelectric effect.
De Broglie's wavelength
Propagation of
de Broglie waves in 1d—real part of the
complex amplitude is blue, imaginary part is green. The probability (shown as the colour
opacity) of finding the particle at a given point
x is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the
curvature decreases, so the amplitude decreases again, and vice versa—the result is an alternating amplitude: a wave. Top:
Plane wave. Bottom:
Wave packet.
In 1924, LouisVictor de Broglie formulated the de Broglie hypothesis, claiming that all matter,^{[22]}^{[23]} not just light, has a wavelike nature; he related wavelength (denoted as λ), and momentum (denoted as p):

\lambda = \frac{h}{p}
This is a generalization of Einstein's equation above, since the momentum of a photon is given by p = \tfrac{E}{c} and the wavelength (in a vacuum) by λ = \tfrac{c}{f}, where c is the speed of light in vacuum.
De Broglie's formula was confirmed three years later for Bell Labs Clinton Joseph Davisson and Lester Halbert Germer guided their beam through a crystalline grid.
De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis. Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.
Heisenberg's uncertainty principle
In his work on formulating quantum mechanics, Werner Heisenberg postulated his uncertainty principle, which states:

\Delta x \Delta p \ge \frac{\hbar}{2}
where

\Delta here indicates standard deviation, a measure of spread or uncertainty;

x and p are a particle's position and linear momentum respectively.

\hbar is the reduced Planck's constant (Planck's constant divided by 2\pi).
Heisenberg originally explained this as a consequence of the process of measuring: Measuring position accurately would disturb momentum and viceversa, offering an example (the "gammaray microscope") that depended crucially on the de Broglie hypothesis. It is now thought, however, that this only partly explains the phenomenon, but that the uncertainty also exists in the particle itself, even before the measurement is made.
In fact, the modern explanation of the uncertainty principle, extending the Copenhagen interpretation first put forward by Bohr and Heisenberg, depends even more centrally on the wave nature of a particle: Just as it is nonsensical to discuss the precise location of a wave on a string, particles do not have perfectly precise positions; likewise, just as it is nonsensical to discuss the wavelength of a "pulse" wave traveling down a string, particles do not have perfectly precise momenta (which corresponds to the inverse of wavelength). Moreover, when position is relatively well defined, the wave is pulselike and has a very illdefined wavelength (and thus momentum). And conversely, when momentum (and thus wavelength) is relatively well defined, the wave looks long and sinusoidal, and therefore it has a very illdefined position.
de Broglie–Bohm theory
Couder experiments,^{[24]} "materializing" the pilot wave model.
De Broglie himself had proposed a pilot wave construct to explain the observed waveparticle duality. In this view, each particle has a welldefined position and momentum, but is guided by a wave function derived from Schrödinger's equation. The pilot wave theory was initially rejected because it generated nonlocal effects when applied to systems involving more than one particle. Nonlocality, however, soon became established as an integral feature of quantum theory (see EPR paradox), and David Bohm extended de Broglie's model to explicitly include it.
In the resulting representation, also called the de Broglie–Bohm theory or Bohmian mechanics,^{[25]} the waveparticle duality vanishes, and explains the wave behaviour as a scattering with wave appearance, because the particle's motion is subject to a guiding equation or quantum potential. "This idea seems to me so natural and simple, to resolve the waveparticle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored",^{[26]} J.S.Bell.
The best illustration of the pilotwave model was given by Couder's 2010 "walking droplets" experiments,^{[27]} demonstrating the pilotwave behaviour in a macroscopic mechanical analog.^{[24]}
Wave behavior of large objects
Since the demonstrations of wavelike properties in photons and electrons, similar experiments have been conducted with neutrons and protons. Among the most famous experiments are those of Estermann and Otto Stern in 1929.^{[28]} Authors of similar recent experiments with atoms and molecules, described below, claim that these larger particles also act like waves.
A dramatic series of experiments emphasizing the action of gravity in relation to wave–particle duality were conducted in the 1970s using the neutron interferometer.^{[29]} Neutrons, one of the components of the atomic nucleus, provide much of the mass of a nucleus and thus of ordinary matter. In the neutron interferometer, they act as quantummechanical waves directly subject to the force of gravity. While the results were not surprising since gravity was known to act on everything, including light (see tests of general relativity and the Pound–Rebka falling photon experiment), the selfinterference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.
In 1999, the diffraction of C_{60} fullerenes by researchers from the University of Vienna was reported.^{[30]} Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 u. The de Broglie wavelength is 2.5 pm, whereas the diameter of the molecule is about 1 nm, about 400 times larger. In 2012, these farfield diffraction experiments could be extended to phthalocyanine molecules and their heavier derivatives, which are composed of 58 and 114 atoms respectively. In these experiments the buildup of such interference patterns could be recorded in real time and with single molecule sensitivity.^{[31]}^{[32]}
In 2003, the Vienna group also demonstrated the wave nature of tetraphenylporphyrin^{[33]}—a flat biodye with an extension of about 2 nm and a mass of 614 u. For this demonstration they employed a nearfield Talbot Lau interferometer.^{[34]}^{[35]} In the same interferometer they also found interference fringes for C_{60}F_{48.}, a fluorinated buckyball with a mass of about 1600 u, composed of 108 atoms.^{[33]} Large molecules are already so complex that they give experimental access to some aspects of the quantumclassical interface, i.e., to certain decoherence mechanisms.^{[36]}^{[37]} In 2011, the interference of molecules as heavy as 6910 u could be demonstrated in a Kapitza–Dirac–Talbot–Lau interferometer. These are the largest objects that so far showed de Broglie matterwave interference.^{[38]} In 2013, the interference of molecules beyond 10,000 u has been demonstrated.^{[39]}
Whether objects heavier than the Planck mass (about the weight of a large bacterium) have a de Broglie wavelength is theoretically unclear and experimentally unreachable; above the Planck mass a particle's Compton wavelength would be smaller than the Planck length and its own Schwarzschild radius, a scale at which current theories of physics may break down or need to be replaced by more general ones.^{[40]}
Recently Couder, Fort, et al. showed^{[41]} that we can use macroscopic oil droplets on a vibrating surface as a model of wave–particle duality—localized droplet creates periodical waves around and interaction with them leads to quantumlike phenomena: interference in doubleslit experiment,^{[42]} unpredictable tunneling^{[43]} (depending in complicated way on practically hidden state of field), orbit quantization^{[44]} (that particle has to 'find a resonance' with field perturbations it creates—after one orbit, its internal phase has to return to the initial state) and Zeeman effect.^{[45]}
Treatment in modern quantum mechanics
Wave–particle duality is deeply embedded into the foundations of quantum mechanics. In the formalism of the theory, all the information about a particle is encoded in its wave function, a complexvalued function roughly analogous to the amplitude of a wave at each point in space. This function evolves according to a differential equation (generically called the Schrödinger equation). For particles with mass this equation has solutions that follow the form of the wave equation. Propagation of such waves leads to wavelike phenomena such as interference and diffraction. Particles without mass, like photons, has no solutions of the Schrödinger equation so have another wave.
The particlelike behavior is most evident due to phenomena associated with measurement in quantum mechanics. Upon measuring the location of the particle, the particle will be forced into a more localized state as given by the uncertainty principle. When viewed through this formalism, the measurement of the wave function will randomly "collapse", or rather "decohere", to a sharply peaked function at some location. For particles with mass the likelihood of detecting the particle at any particular location is equal to the squared amplitude of the wave function there. The measurement will return a welldefined position, (subject to uncertainty), a property traditionally associated with particles. It is important to note that a measurement is only a particular type of interaction where some data is recorded and the measured quantity is forced into a particular eigenstate. The act of measurement is therefore not fundamentally different from any other interaction.
Following the development of quantum field theory the ambiguity disappeared. The field permits solutions that follow the wave equation, which are referred to as the wave functions. The term particle is used to label the irreducible representations of the Lorentz group that are permitted by the field. An interaction as in a Feynman diagram is accepted as a calculationally convenient approximation where the outgoing legs are known to be simplifications of the propagation and the internal lines are for some order in an expansion of the field interaction. Since the field is nonlocal and quantized, the phenomena which previously were thought of as paradoxes are explained. Within the limits of the waveparticle duality the quantum field theory gives the same results.
Visualization
There are two ways to visualize the waveparticle behaviour: by the "standard model", described below; and by the Broglie–Bohm model, where no duality is perceived.
Below is an illustration of wave–particle duality as it relates to De Broglie's hypothesis and Heisenberg's uncertainty principle (above), in terms of the position and momentum space wavefunctions for one spinless particle with mass in one dimension. These wavefunctions are Fourier transforms of each other.
The more localized the positionspace wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentumspace wavefunction is less localized so the possible momentum components the particle could have are more widespread.
Conversely the more localized the momentumspace wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the positionspace wavefunction, so the position coordinates the particle could occupy are more widespread.
Position
x and momentum
p wavefunctions corresponding to quantum particles. The colour opacity (%) of the particles corresponds to the probability density of finding the particle with position
x or momentum component
p.
Top: If wavelength
λ is unknown, so are momentum
p, wavevector
k and energy
E (de Broglie relations). As the particle is more localized in position space, Δ
x is smaller than for Δ
p_{x}.
Bottom: If λ is known, so are p, k, and E. As the particle is more localized in momentum space, Δp is smaller than for Δx.
Alternative views
Wave–particle duality is an ongoing conundrum in modern physics. Most physicists accept waveparticle duality as the best explanation for a broad range of observed phenomena; however, it is not without controversy. Alternative views are also presented here. These views are not generally accepted by mainstream physics, but serve as a basis for valuable discussion within the community.
Bothparticleandwave view
The pilot wave model, originally developed by Louis de Broglie and further developed by David Bohm into the hidden variable theory proposes that there is no duality, but rather a system exhibits both particle properties and wave properties simultaneously, and particles are guided, in a deterministic fashion, by the pilot wave (or its "quantum potential") which will direct them to areas of constructive interference in preference to areas of destructive interference. This idea is held by a significant minority within the physics community.^{[46]}
At least one physicist considers the "waveduality" a misnomer, as L. Ballentine, Quantum Mechanics, A Modern Development, p. 4, explains:
When first discovered, particle diffraction was a source of great puzzlement. Are "particles" really "waves?" In the early experiments, the diffraction patterns were detected holistically by means of a photographic plate, which could not detect individual particles. As a result, the notion grew that particle and wave properties were mutually incompatible, or complementary, in the sense that different measurement apparatuses would be required to observe them. That idea, however, was only an unfortunate generalization from a technological limitation. Today it is possible to detect the arrival of individual electrons, and to see the diffraction pattern emerge as a statistical pattern made up of many small spots (Tonomura et al., 1989). Evidently, quantum particles are indeed particles, but whose behaviour is very different from classical physics would have us to expect.
The Afshar experiment^{[47]} (2007) has demonstrated that it is possible to simultaneously observe both wave and particle properties of photons.
Waveonly view
At least one scientist proposes that the duality can be replaced by a "waveonly" view. In his book Collective Electrodynamics: Quantum Foundations of Electromagnetism (2000), Carver Mead purports to analyze the behavior of electrons and photons purely in terms of electron wave functions, and attributes the apparent particlelike behavior to quantization effects and eigenstates. According to reviewer David Haddon:^{[48]}
Mead has cut the Gordian knot of quantum complementarity. He claims that atoms, with their neutrons, protons, and electrons, are not particles at all but pure waves of matter. Mead cites as the gross evidence of the exclusively wave nature of both light and matter the discovery between 1933 and 1996 of ten examples of pure wave phenomena, including the ubiquitous laser of CD players, the selfpropagating electrical currents of superconductors, and the Bose–Einstein condensate of atoms.
Albert Einstein, who, in his search for a Unified Field Theory, did not accept waveparticle duality, wrote:^{[49]}
This double nature of radiation (and of material corpuscles)...has been interpreted by quantummechanics in an ingenious and amazingly successful fashion. This interpretation...appears to me as only a temporary way out...
The manyworlds interpretation (MWI) is sometimes presented as a wavesonly theory, including by its originator, Hugh Everett who referred to MWI as "the wave interpretation".^{[50]}
The Three Wave Hypothesis of R. Horodecki relates the particle to wave.^{[51]}^{[52]} The hypothesis implies that a massive particle is an intrinsically spatially as well as temporally extended wave phenomenon by a nonlinear law.
Neitherwavenorparticle view
It has been argued that there are never exact particles or waves, but only some compromise or intermediate between them. One consideration is that zerodimensional mathematical points cannot be observed. Another is that the formal representation of such points, the Kronecker delta function is unphysical, because it cannot be normalized. Parallel arguments apply to pure wave states. Roger Penrose states:^{[53]}
"Such 'position states' are idealized wavefunctions in the opposite sense from the momentum states. Whereas the momentum states are infinitely spread out, the position states are infinitely concentrated. Neither is normalizable [...]."
Relational approach to wave–particle duality
Relational quantum mechanics is developed which regards the detection event as establishing a relationship between the quantized field and the detector. The inherent ambiguity associated with applying Heisenberg's uncertainty principle and thus wave–particle duality is subsequently avoided.^{[54]}
Applications
Although it is difficult to draw a line separating wave–particle duality from the rest of quantum mechanics, it is nevertheless possible to list some applications of this basic idea.

Wave–particle duality is exploited in electron microscopy, where the small wavelengths associated with the electron can be used to view objects much smaller than what is visible using visible light.

Similarly, neutron diffraction uses neutrons with a wavelength of about 0.1 nm, the typical spacing of atoms in a solid, to determine the structure of solids.
See also
Notes and references

^ Harrison, David (2002). "Complementarity and the Copenhagen Interpretation of Quantum Mechanics". UPSCALE. Dept. of Physics, U. of Toronto. Retrieved 20080621.

^ ^{a} ^{b} Schrödinger, E. (1928). Wave mechanics, pp. 185–206 of Électrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique, tenu à Bruxelles du 24 au 29 Octobre 1927, sous les Auspices de l'Institut International de Physique Solvay, GauthierVillars, Paris, pp. 185–186; translation at p. 447 of Bacciagaluppi, G., Valentini, A. (2009), Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, Cambridge UK, ISBN 9780521814218.

^ Bransden, B.H., Joachain, C.J. (1989/2000). Quantum Mechanics, second edition, Pearson, Prentice Hall, Harlow UK, ISBN 9780582356917, p. 760.

^ Duane, W. (1923). (5): 158–1649 Proc. Natl. Acad. Sci.The transfer in quanta of radiation momentum to matter, .

^ Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman and Sons, London, pp. 19–22.

^ Heisenberg, W. (1930). The Physical Principles of the Quantum Theory, translated by C. Eckart and F.C. Hoyt, University of Chicago Press, Chicago, pp. 77–78.

^ Heisenberg, W., (1967). Quantum theory and its interpretation, quoted on p. 56 by eds. J.A. Wheeler, W.H. Zurek, (1983), Quantum Theory and Measurement, Princeton University Press, Princeton NJ, from ed. S. Rozental, Niels Bohr: his Life and Work as seen by his Friends and Colleagues, North Holland, Amsterdam.

^ Kumar, Manjit (2011). Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality (Reprint edition ed.). W. W. Norton & Company.

^ Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, : 580–590121 Supplement April 14 1928, Nature.

^ Camilleri, K. (2009). Heisenberg and the Interpretation of Quantum Mechanics: the Physicist as Philosopher, Cambridge University Press, Cambridge UK, ISBN 9780521884846.

^ Preparata, G. (2002). An Introduction to a Realistic Quantum Physics, World Scientific, River Edge NJ, ISBN 9812391767.

^ Walter Greiner (2001). Quantum Mechanics: An Introduction. Springer.

^ R. Eisberg and R. Resnick (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). John Wiley & Sons. pp. 59–60.

^ Nathaniel Page Stites, M.A./M.S. "Light I: Particle or Wave?," Visionlearning Vol. PHY1 (3), 2005. http://www.visionlearning.com/library/module_viewer.php?mid=132

^ Young, Thomas (1804). "Bakerian Lecture: Experiments and calculations relative to physical optics". Philosophical Transactions of the Royal Society 94: 1–16.

^ Thomas Young: The Double Slit Experiment

^

^ http://dx.doi.org/10.1119/1.1737397

^ "light", The Columbia Encyclopedia, Sixth Edition. 2001–05.

^ Buchwald, Jed (1989). The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century. Chicago: University of Chicago Press.

^ Zhang, Q (1996). "Intensity dependence of the photoelectric effect induced by a circularly polarized laser beam". Physics Letters A 216 (15): 125.

^ Donald H Menzel, "Fundamental formulas of Physics", volume 1, page 153; Gives the de Broglie wavelengths for composite particles such as protons and neutrons.

^ Brian Greene, The Elegant Universe, page 104 "all matter has a wavelike character"

^ ^{a} ^{b} See this Science Channel production (Season II, Episode VI "How Does The Universe Work?"), presented by Morgan Freeman, https://www.youtube.com/watch?v=W9yWv5dqSKk

^ Bohmian Mechanics, Stanford Encyclopedia of Philosophy.

^ Bell, J. S., "Speakable and Unspeakable in Quantum Mechanics", Cambridge: Cambridge University Press, 1987.

^ Y. Couder, A. Boudaoud, S. Protière, Julien Moukhtar, E. Fort: Walking droplets: a form of waveparticle duality at macroscopic level? , doi:10.1051/epn/2010101, (PDF)

^ Estermann, I.; Stern O. (1930). "Beugung von Molekularstrahlen". Zeitschrift für Physik 61 (12): 95–125.

^ R. Colella, A. W. Overhauser and S. A. Werner, Observation of Gravitationally Induced Quantum Interference, Phys. Rev. Lett. 34, 1472–1474 (1975).

^ Arndt, Markus; O. Nairz,

^ Juffmann, Thomas et al (25 March 2012). "Realtime singlemolecule imaging of quantum interference".

^ Quantumnanovienna. "Single molecules in a quantum interference movie". Retrieved 20120421.

^ ^{a} ^{b} Hackermüller, Lucia; Stefan Uttenthaler, Klaus Hornberger, Elisabeth Reiger, Björn Brezger, Anton Zeilinger and Markus Arndt (2003). "The wave nature of biomolecules and fluorofullerenes". Phys. Rev. Lett. 91 (9): 090408.

^ Clauser, John F.; S. Li (1994). "Talbot von Lau interefometry with cold slow potassium atoms.". Phys. Rev. A 49 (4): R2213–17.

^ Brezger, Björn; Lucia Hackermüller, Stefan Uttenthaler, Julia Petschinka, Markus Arndt and Anton Zeilinger (2002). "Matterwave interferometer for large molecules". Phys. Rev. Lett. 88 (10): 100404.

^ Hornberger, Klaus; Stefan Uttenthaler,Björn Brezger, Lucia Hackermüller, Markus Arndt and Anton Zeilinger (2003). "Observation of Collisional Decoherence in Interferometry". Phys. Rev. Lett. 90 (16): 160401.

^ Hackermüller, Lucia; Klaus Hornberger, Björn Brezger, Anton Zeilinger and Markus Arndt (2004). "Decoherence of matter waves by thermal emission of radiation". Nature 427 (6976): 711–714.

^ Gerlich, Stefan; et al. (2011). "Quantum interference of large organic molecules".

^ Eibenberger, S.; Gerlich, S.; Arndt, M.; Mayor, M.; Tüxen, J. (2013). "Matter–wave interference of particles selected from a molecular library with masses exceeding 10 000 amu". Physical Chemistry Chemical Physics 15 (35): 14696–14700.

^ Peter Gabriel Bergmann, The Riddle of Gravitation, Courier Dover Publications, 1993 ISBN 0486273784 online

^ http://www.youtube.com/watch?v=W9yWv5dqSKk  You Tube video  Yves Couder Explains Wave/Particle Duality via Silicon Droplets

^ Y. Couder, E. Fort, SingleParticle Diffraction and Interference at a Macroscopic Scale, PRL 97, 154101 (2006) online

^ A. Eddi, E. Fort, F. Moisy, Y. Couder, Unpredictable Tunneling of a Classical Wave–Particle Association, PRL 102, 240401 (2009)

^ E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, Y. Couder, Pathmemory induced quantization of classical orbits, PNAS October 12, 2010 vol. 107 no. 41 1751517520

^ http://prl.aps.org/abstract/PRL/v108/i26/e264503  Level Splitting at Macroscopic Scale

^ (Buchanan pp. 29–31)

^ Afshar S.S. et al: Paradox in Wave Particle Duality. Found. Phys. 37, 295 (2007) http://arxiv.org/abs/quantph/0702188 arXiv:quantph/0702188

^ David Haddon. "Recovering Rational Science". Touchstone. Retrieved 20070912.

^ Paul Arthur Schilpp, ed, Albert Einstein: PhilosopherScientist, Open Court (1949), ISBN 0875481337, p 51.

^ See section VI(e) of Everett's thesis: The Theory of the Universal Wave Function, in Bryce Seligman DeWitt, R. Neill Graham, eds, The ManyWorlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 069108131X, pp 3–140.

^ Horodecki, R. (1981). "De broglie wave and its dual wave".

^ Horodecki, R. (1983). "Superluminal singular dual wave". Lett. Novo Cimento 38: 509–511.

^

^ http://www.quantumrelativity.org/QuantumRelativity.pdf See Q. Zheng and T. Kobayashi, Quantum Optics as a Relativistic Theory of Light; Physics Essays 9 (1996) 447. Annual Report, Department of Physics, School of Science, University of Tokyo (1992) 240.
External links

Animation, applications and research linked to the waveparticle duality and other basic quantum phenomena (Université Paris Sud)

H. Nikolic. "Quantum mechanics: Myths and facts". arXiv:quantph/0609163.

Young & Geller. "College Physics".

B. Crowell. "Light as a Particle" (Web page). Retrieved December 10, 2006.

E.H. Carlson, Wave–Particle Duality: Light on Project PHYSNET

R. Nave. "Wave–Particle Duality" (Web page). HyperPhysics. Georgia State University, Department of Physics and Astronomy. Retrieved December 12, 2005.

Juffmann, Thomas et al (25 March 2012). "Realtime singlemolecule imaging of quantum interference".
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